62 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA E/M is also Hausdorff (12.11.2), the canonical mapping of N onto E/M, which is linear and bijective, is bicontinuous by (ii) because both N and E/M are finite-dimensional. Next, consider (i). If/is continuous it is clear that H =f~*(G) is closed. The converse follows from (iv): if f(a) = 1, then the line D = R<z (resp. D = Co) is supplementary to H and therefore a topological supplement; hence the mapping x\~»f(x)a is continuous, and by (ii) this implies the continuity of/. Finally, to prove (iii), assume first that M = {0}. Then F is a locally com- pact metrizable subgroup by (ii), and we can apply (12.9.6). In the general case, consider E/M and apply the characterization of closed sets in E/M (12.10). The result (ii) above may also be expressed as follows: on a finite-dimen- sional vector space (over R or C), there is a unique Hausdorff topology com- patible with the vector space structure (every endomorphism of the vector space R" (resp. C1) being necessarily continuous with respect to the product topology). This topology is said to be canonical. (12.13.3) Let E1? E2 be two metrizable topological vector spaces, and assume that E2 is complete. Let Fx (resp. F2) be a dense vector subspace o/Ex (resp. E2). Then every continuous linear mapping u: Ft -> F2 has a unique extension to a continuous linear mapping u: E1 -> E2. If in addition E1 is complete and u is an isomorphism (of topological vector spaces) of „1 onto F2, then w is an isomorphism (of topological vector spaces) ofE± onto E2 . In view of (12.9.4), we have only to show that u is linear, i.e., that iZ(/bc) = Aw(x) for all x e E1 and all scalars A; but this is a consequence of the principle of extension of identities (3.15.2). 14. LOCALLY CONVEX SPACES Let E be a vector space over R or C. A seminorm on E is a mapping/? of E into R which satisfies conditions (I), (III), and (IV) of (5.1). A norm is there- fore a seminorm/? such that/?(*) ^ 0 for all x ^ 0 in E. If/? is a seminorm on E, then d(x, y) = p(x - y} is a pseudo-distance on E which is translation- invariant and such that d(Ax, ky) = |A| d(x, y) for all scalars 1 (12.14.1) Let (px)a e, be a family ofseminorms on a vector space E. If the upper envelope p~ sup /?a is finite on E, then it is a seminorm.r assertion (12.2.1).